Question

chứng minh 1/3+1/3^2+1/3^3+...+1/3^2011+1/3^2012 < 1/2

Answer 1

Đặt A = 1/3 + 1/3^2 + 1/3^3 + ... + 1/3^2011 + 1/3^2012

3A = 1 + 1/3 + 1/3^2 + ... + 1/3^2010 + 1/3^2011

3A - A = ( 1 + 1/3 + 1/3^2 + ... + 1/3^2010 + 1/3^2011) - ( 1/3 + 1/3^2 + 1/3^3 + ... + 1/3^2011 + 1/3^2012)

Answer 2

A= 1/3+1/3^2+1/3^3+...+1/3^2011+1/3^2012 

1/3.A= 1/3^2+1/3^3+1/3^4+...+1/3^2012+1/3^2013

=> 1/3.A-A=-2/3.A = (1/3^2+1/3^3+1/3^4+...+1/3^2012+1/3^2013) - ( 1/3+1/3^2+1/3^3+...+1/3^2011+1/3^2012 )

=> -2/3.A= 1/3^2013 +1/3

=> A= (1/3^2013+1/3) : -2/3

Ta được A < 1/2 

:D